We stress that if \(p=1\)  (purely Gaussian patterns), assuming \(m\ne 0\)  in Eq. (REF ), yields \(1=\beta (1-q)\)  which causes Eq. (REF ) to be ill defined. A well defined theory is recovered only if we had considered \(\alpha =0\)  in the beginning, that is, \(P\)  finite, not scaling with \(N\) . Indeed, in this case there is no need to introduce the variable \(r\)  which, in fact, represents the random overlaps with the mismatched patterns, acting as an overall Gaussian noise. A description of the Gaussian Hopfield model at \(\alpha =0\)  can be found in  [1]}.
